Sensing and edge artificial intelligence (AI) are envisioned as two essential and interconnected functions in sixth-generation (6G) mobile networks. On the one hand, sensing-empowered applications rely on powerful AI models to extract features and understand semantics from ubiquitous wireless sensors. On the other hand, the massive amount of sensory data serves as the fuel to continuously refine edge AI models. This deep integration of sensing and edge AI has given rise to a new task-oriented paradigm known as integrated sensing and edge AI (ISEA), which features a holistic design approach to communication, AI computation, and sensing for optimal sensing-task performance. In this article, we present a comprehensive survey for ISEA. We first provide technical preliminaries for sensing, edge AI, and new communication paradigms in ISEA. Then, we study several use cases of ISEA to demonstrate its practical relevance and introduce current standardization and industrial progress. Next, the design principles, metrics, tradeoffs, and architectures of ISEA are established, followed by a thorough overview of ISEA techniques, including digital air interface, over-the-air computation, and advanced signal processing. Its interplay with various 6G advancements, e.g., new physical-layer and networking techniques, are presented. Finally, we present future research opportunities in ISEA, including the integration of foundation models, convergence of ISEA and integrated sensing and communications (ISAC), and ultra-low-latency ISEA.